Infinitary Normalization

In infinitary orthogonal first-order term rewriting the properties confluence (CR), Uniqueness of Normal forms (UN), Parallel Moves Lemma (PML) have been generalized to their infinitary versions CR∞, UN∞, PML∞, and so on. Several relations between these properties have been established in the literature. Generalization of the termination properties, Strong Normalization (SN) and Weak Normalization (WN) to SN∞ and WN∞ is less straightforward. We present and explain the definitions of these infinitary normalization notions, and establish that as a global property of orthogonal TRSs they coincide, so at that level there is just one notion of infinitary normalization. Locally, at the level of individual terms, the notions are still different. In the setting of orthogonal term rewriting we also provide an elementary proof of UN∞, the infinitary Unique Normal form property.